The Belle II experiment is under construction at the KEK laboratory in Japan. Belle II will study $e^+e^-$ collisions at or near the $\Upsilon(4S)$ resonance with the goal of collecting 50 ab$^{-1}$ of data, which is a large increase over that recorded by the Belle and BaBar experiments. This data will provide a large sample of charm meson decays. In this report we present the expected sensitivity of Belle II for measuring time-dependent mixing and $CP$ violation in the $D^0$-$\overline{D}{}^0$ system. We focus on measurements of
$D^0\rightarrow K^+\pi^-$, $D^0\rightarrow K^+\pi^-\pi^0$, and $D^0\rightarrow K^0_S\pi^+\pi^-$ decays.